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Proceedings of the Cambridge Philosophical Society, Mathematical and 

physical sciences. 

Cambridge [etc.]Cambridge Philosophical Society. 
http://www.biodiversitylibrary.org/bibliography/44060 



V. 15 (1908-10): http://www.biodiversitylibrary.org/item/97262 
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Mr BatemaUy Solution of a s 




I 




equations, etc. 423 



The solution of a system of differential 




occurring 



in the theory of radio-active transformations. By H. Bateman, 
M,A., Trinity College. 

[Read 21 February 1910.] 

1. It has been shown by Prof. Rutherford * that the amounts 
of the primary substance and the different products in a given 

• . rh ^« *■ ii T*ij1 i r* 



quantity of radio-active matter vary according to the system of 



d 




equations, 



dt 

d^ 

dt 



\P 



\iJr — A»2y 



-— = \^Q — Xjit 



(1), 



d^ 
dt 



A? a — \a1 



» • 



; 



where P, Q, R,S,T,... denote the number of atoms of the primary 
substance and successive products which are present at time t. 

Prof Rutherford has worked out the various cases in which 
there are only two products in addition to the primary substance, 
and it looks at first sight as if the results may be extended to any 
number of products without much labour. 

Unfortunately the straightforward method is unsymmetrical 
and laborious, and as the results of the calculations are needed in 
some of the researches which are being carried on in radio-activity 
the author has thought it worth while to publish a simple and 
symmetrical method of obtaining the required formulae. 

2. Let us introduce a set of auxiliary, quantities p (^), q (a?), 



r {a))y 



depending on a variable ^ and connected with the 



quantities P{t)y Q(t), R{t\ .••by the equations, 



p{x) 




00 



00 



e-''^P(t)dt, q{x) 











e~''^Q{t)dt (2). 



It is easily seen that 




00 



e 



3/5 







dP 
dt 



dt 



A 00 

P(0) + ^ e-^^Pif)dt (3) 

JO 



? 



Pis-^xp 



J 



* 



Radio-activity, 2nd edition, p. 331. 




Ml' Bateman, Solution of a system of differential equations 



where p is written for p (w), and P^ for P (0), the initial value 
of P (t). 

Multiplying equations (1) by e~^\ and integrating from to 
00 with regard to t we obtain the system of equations 



XP X^Q 

Qo 

Sn 



a-q 



/>*^'* 






-\p 

\ip — X^q 

^2q 



A4.9 






from which the values of jj, q, r may be obtained at once. 



If Q.^R 







S^ 



m m 



present initially, we have 



0; i.e. if there is only one substance 



P 



X + Xi 



<1 



{x + Xi) {x + Xo) ' 



/v» 



Aj A2 JTq 



{x + Xj) (^ + X2) {x + X3) ' 



and for the ni\\ product 



V {x) 



Ax Ag • • • A^_i X"^, 







(^ + Xi) (^ + Xo) . . . {x+ Xn) * 






Putting this into partial fractions, we have 



v{x) 



a 



+ 



a 



^^t^^^ T I. 



^ + Xi ^ + X 



1 • • • 



c 



n 



X + Ayj 



where 



c 



Aj A2 • • • Ajj 2 -i 







(X 



Xl) (X; 



Xl) . . . (X;i — Xj) 



Go 



/vj /V^ • • • "^ji 1 -^ 



y 



\\i — X2) (Xg — Xo) ... (X 



il 



Xo) 



(6). 



• • « * 



6 uC, 



To obtain the corresponding function N{t) we must solve the 



integral equation 



V (x) 




00 



e-^^ N {t) dt 







Now it has been shown by Lerch^ that there is only one 
continuous function N (t) which will yield a given function v (x) ; 
hence if we can find a function which satisfies this condition it 
will be the solution of our problem. It is clear, however, that 



1 



^ + X 




00 



^-xt ^ ^-Kt ^l . 







hence the above value of v {x) is obtained by taking 

N {t) ^ c^e'^^^^ ^^ c^e^^^^ + ... Cne'^^'^ 
e the constants have the values given by (6). 



(7), 




* Acta Mathematical 1903, p. 339. 



occurring in the theory of radio-active transformations. 




In the case when Q (0), R (0), . . . are not zero, we have 



P 



•* 



iC+Xi' 



? 



\,P 







+ 



Qo 



{X + A-i) (il? + Xa) ^ + Xo 



r 



^1^2-^ 







+ 



A2V0 



+ 



-Bo 



•(8), 



(x + \i) (a; + Xa) («^ + 5^3) (« + ^2) («? + ^3) (« + ^3) 



etc. 



and we may obtain the values of P, Q, R by expressing these 
quantities in partial fractions as before. 

The complete solution for the case of a primary substance P 
and three products Q, R, S is 



P = Poe-^'^ Q 



\ 



X 



\ 



Png-^'^ + 



^''^'- + Qo) e-'^* 



R 



A,^ A^2 -^0 



(Xo — Xi) (X,3 — Xj) 



6 



Aif 



+ 



Xi — X 

A.2 A.2 -to 



+ 



X2 Vo 



(Xj — X2) (X3 — X2) X 



A^9 



e 



At^iyjj 



+ 



\l\2Jr 







+ 



X2Q 







(Xj — X3) (Xg — X3) X.2 — X 



+ R 








e 



Aqt 



8 



KiK^*^^-^ Q 



(Xo X 1 ) (X; 



\i) (X. 



Xi) 



e 



AjT 



+ 



* 

Xi Xg X3 X 







-. + 



X'2 X3 t^Q 



(Xi — X2) (Xo — X2) (X4 — Xo) (X3 — X2) (X4 — Xy)^ 



e 



\of 



+ 



XiX2X3-r 







m'^^ t ■ 1 I 



+ 



A-2 A3 (^Q 



_(Xi — X3) (X2 — X3) (X4 — X3) (X2 — X3) (X 



"^3) 



+ 



^4 — ^1 



Q-\st + 



Xt Xo Xct ± 

X tkj MJ 







_(Xl — X4) (X2 — X4) (X; 



\) 



+ 



Xo X3 {^Q 






(X2 — X4) (X3 — X4) X3 — X4 



J 



e 



x,t 



The solution may evidently be obtained by superposing the 
solutions of the cases in which the initial values of P, Q, R, S are 
given by 



(1) P(0) = Po, 

(2) P(0) = 0, 

(3) P(0) = 0, 

(4) P(0) = 0. 



0, R(0) 



0, 



S{0) 



0. 



QiO) 
Q{0) 

Q(0) = 0, R(0) = 0, 5(0) = ^0. 



Q„ R{0) = 0, S{0) = 0. 
0, R(0) = Ro, S{0) = 0. 



The method is perfectly general, and the corresponding 
formulae for the case of ?i— 1 products may be written down at 
once by using (6). 

The general formula covers all the four cases {Radio-activity, 
pp. 331 — 337). For instance in Case 2 when initially there is 
radio-active equilibrium, we have 



n 







X,P 







Xov^o — K^l\,(^ — A^>O0, 



426 Mr Bateman, Solution of a system of differential equations 



The solutions are then 



P 



n 







\ 



e 



\,t 



Q 



n 







A- 



\ 



e 



^i^ -f- 



n^Xj 



-^^f^m 



R 



Xo7? 



2"0 



(Xo — Xj) (X. 



K) 



i 



Xo (X^ — Xo) 

Xi?lo 



e 



Agf 



: e 



Kot 



+ 



{\l — X2) (X3 — Xo) 

Xj X2 "Hq 
^^ (^1 **" Xo) (X2 •— X3) 



e 



Agf 



S 



Xo Xo 71 







(X2 — Xi) (Xo — Xi) (X4 — Xj) 



e 



Kt 



+ 



Xi Xo V 

A O 







+ 



(Xi — X2) (X3 — X2) (X4 — Xg) 



e 



AciV 



(Xi — Xg) (X2 — X3) (X. 



^) 



e 



Aqt 



+ 



XiX2X3?l 







(Xi — X4) (Xj 



X4) (Xg 



X4) 



e' 



•^4,* 



The solution for the ease of ?i— 1 products is given by 



i\^ = 2a 



Ov w 



Ai»t 



^j 



where the constants Cr are obtained by expressing 



1 2 3 • • • A/ 



?i— 1 



00 (00 -\- Xi) (^' + Xg) ... (^ + X^i) 



in partial fractions. 

The method by which the solution of the system of differen- 
tial equations has been obtained is really of very wide application 
and may be employed to solve problems depending on a partial 
differential equation of the form 



f(1 1 i 

provided the initial value of V is known. 



dV 

dt 



V, 



For if we put 



QO 



u(s)= I e-'^ V{t)dt, 

Jo 



su (s) — Vq 




00 







dV 

e'^'^-^dt, 

ot 



it appears that it(s) satisfies the partial differential equation 



f( 



d 



d 



d 



\ 



\da) ' dy' dz^ 



U'\-su+ Fo = (9). 



Further, if V satisfies some linear boundary condition which 
is independent of t the function ii will generally satisfy the same 
boundary condition. This function {%i) must be obtained from the 



occurring in the theory of radio-active transformation^. 



427' 



differential equation (10) which is simpler than (9), inasmuch as 
it depends upon fewer independent variables. 

Tn many cases the solution of the integral equation 



^t (s) 




00 



e-'^ V{t) dt 







may be calculated by means of the inversion formula 



* 



V{t) 



1 



27ri 




e*^ u (?) d^, 



e 



where c is a contour which starts at — oo at a point below the 



real 



(r) 



returns to - x at a point above the real axis, as in the figure. 




The conditions to be satisfied by ii (^) 



that this 



inversion formula may be applicable have not yet been expressed 
in a concise form. 

The formula may be used to obtain the solution of a problem 
in the conduction of heat when we require a solution of 

d'V dV 



dx^ 



dt ' 



t • « 



which satisfies the boundary conditions 

V=^ when ^ = and a) = a^ 

V =f(,x) when i = 0, 

The solution found in this way is identical with the one given 
in Carslaw's Fourier s Series and Integrals, p. 383, 



* A particular case of this formula has been given by Pincherle, Bologna 
Memoirs, 10 (8), 1887.